80.4.3 problem 9

Internal problem ID [21220]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 4. Existence and uniqueness for systems and higher order. Excercise 4.3 at page 76
Problem number : 9
Date solved : Thursday, October 02, 2025 at 07:27:00 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ x^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 6

ode:=diff(diff(x(t),t),t)+x(t) = 0; 
ic:=[x(0) = 0, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \sin \left (t \right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 7

ode=D[x[t],{t,2}]+x[t]==0; 
ic={x[0]==0,Derivative[1][x][0] ==1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \sin (t) \end{align*}

✓ Sympy. Time used: 0.029 (sec). Leaf size: 5

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \sin {\left (t \right )} \]